The ith row of A represents a ±1-valued “square wave” function of j with halfperiod i. Thus xA, where x is a (row) N -vector, gives a picture of the waveform that is a linear combination of square waves with weights in x. Similarly yA finds weights so that a given waveform y is a weighted linear combination of these square waves. This matrix has a surprisingly simple inverse. More generally, we...

One approach to computing a square root of a matrix A is to apply Newton's method to the quadratic matrix equation F( X) = X2 A =0. Two widely-quoted matrix square root iterations obtained by rewriting this Newton iteration are shown to have excellent mathematical convergence properties. However, by means of a perturbation analysis and supportive numerical examples, it is shown that these simpl...

j=1 (−1) +j · ai∗j · det(M i∗j). (1) Besides det(A), we may also denote the determinant of A as |A|. Henceforth, if we apply (1) to compute det(A), we say that we expand A by row i∗. It is important to note that the value of det(A) does not depend on the choice of i∗. We omit a proof of this fact, but illustrate it in the following examples. Example 1 (Second-Order Determinants). In general, if...

Two-particle scattering amplitudes for integrable relativistic quantum field theory in 1+1 dimensions can normally have at most singularities of poles and zeros along the imaginary axis in the complex rapidity plane. It has been supposed that single particle amplitudes of the exact boundary reflection matrix exhibit the same structure. In this paper, single particle amplitudes of the exact boun...